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Non-Gaussian fluctuations of randomly trapped random walks

Adam Bowditch

TL;DR

This work analyzes one-dimensional biased randomly trapped random walks with trap times of infinite variance, establishing functional limit theorems for the clock process and the position under sub-ballistic and ballistic regimes. The clock process S_t, after appropriate scaling, converges to a Lévy process, with a stable-subordinator limit under strong subsequential stability, and the walk position converges to a transform of its inverse, revealing non-Gaussian fluctuations. In the ballistic regime, the study shows joint convergence of hitting times and positions to Lévy processes, with α-stable limits along subsequences under a Laplace-transform framework. The results are applied to biased walks on subcritical Galton–Watson trees conditioned to survive, yielding a phase diagram of fluctuation regimes and highlighting lattice effects that prevent full J1 convergence; the work also clarifies the limitations of extending annealed results to quenched settings in low dimensions. This advances understanding of trapping phenomena in low-dimensional random walks and informs related models on random graphs and trees.

Abstract

In this paper we consider the one-dimensional, biased, randomly trapped random walk when the trapping times have infinite variance. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton-Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

Non-Gaussian fluctuations of randomly trapped random walks

TL;DR

This work analyzes one-dimensional biased randomly trapped random walks with trap times of infinite variance, establishing functional limit theorems for the clock process and the position under sub-ballistic and ballistic regimes. The clock process S_t, after appropriate scaling, converges to a Lévy process, with a stable-subordinator limit under strong subsequential stability, and the walk position converges to a transform of its inverse, revealing non-Gaussian fluctuations. In the ballistic regime, the study shows joint convergence of hitting times and positions to Lévy processes, with α-stable limits along subsequences under a Laplace-transform framework. The results are applied to biased walks on subcritical Galton–Watson trees conditioned to survive, yielding a phase diagram of fluctuation regimes and highlighting lattice effects that prevent full J1 convergence; the work also clarifies the limitations of extending annealed results to quenched settings in low dimensions. This advances understanding of trapping phenomena in low-dimensional random walks and informs related models on random graphs and trees.

Abstract

In this paper we consider the one-dimensional, biased, randomly trapped random walk when the trapping times have infinite variance. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton-Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

Paper Structure

This paper contains 8 sections, 19 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sub-ballistic regime
  3. Ballistic regime
  4. Counterexample for convergence in the Skorohod topology
  5. Random walks on Galton-Watson trees
  6. The walk on the tree as a randomly trapped random walk
  7. A decomposition of the trapping times
  8. Subsequential convergence of the Laplace transforms

Key Result

Theorem 1

Let $\beta>1$, $\alpha\in(0,1)$ and $f:\mathbb{N}\times\mathbb{R}_+\rightarrow \mathbb{R}$ be such that $f(1,\lambda)\asymp\lambda^\alpha$. Suppose there exists $n_k\nearrow \infty$, $a_n=n^{1/\alpha}L(n)$ for some $L$ slowly varying such that for any $l \in\mathbb{N}$ and $\lambda>0$ we have as $k\rightarrow \infty$. Then, as $k\rightarrow \infty$, $(S_{n_kt}/a_{n_k},X_{a_{n_k}t}/n_k)_{t\geq 0}$

Figures (2)

  • Figure 1: A sample subcritical GW-tree conditioned to survive $\mathcal{T}$ with the backbone $\mathcal{Y}$ represented by solid lines and the buds and traps connected by dashed lines.
  • Figure 2: Phase diagram depicting the escape regimes of the biased random walk on the subcritical Galton-Watson tree for different combinations of mean and bias. In region A the walk is recurrent. In region B the walk is ballistic and satisfies a functional central limit theorem. In region C the walk is ballistic and has fluctuations of order $n^{1/\gamma}$. In region D the walk is subballistic and the distance reached by the walker by time $n$ is of order $n^\gamma$ (meaning level $n$ is first reached around time $n^{1/\gamma}$).

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof : Proof of Theorem \ref{['t:Sstab']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 25 more